varun289 wrote:If j > 1, is integer j a prime number?

(1) When j is divided by 3, the remainder is 1.
(2) When j is divided by 2, the remainder is 1.

One simple thing about prime number is they are rarely predictable and there is no formula to generate a prime number. A prime number may be represented in a particular form but that particular form will not always generate prime numbers.

So, whenever you come across this type of problem, combine both statements together and try to find contradiction. In fact, with practice you'll happen to know that answer to this particular problem is definitely E. Find a contradiction just to be more confident.

Here, if we combine both statements, it simply means when j is divided by 6 it leaves a remainder of 1. So, j is one more than a multiple of 6.

7 = 6*1 + 1 ---> 7 is prime
25 = 6*4 + 1 ---> 25 is not prime

We have our contradiction.
So, both statements together is also not sufficient.

The correct answer is E.

Go through this discussion made by Anurag on a problem testing the same concept (please scroll down and read his follow-on posts also) >> https://www.beatthegmat.com/num-prp-t74492.html#335738

varun289 wrote:If j > 1, is integer j a prime number?

(1) When j is divided by 3, the remainder is 1.

(2) When j is divided by 2, the remainder is 1.

When it comes to remainders, we have a nice rule that says:

If N divided by D, leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.

For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

Okay, now onto the question.

Target question: Is integer j a prime number?

Statement 1: When j is divided by 3, the remainder is 1.
Possible values of j: 4, 7, 10, 13, 16, 19, 22, 25...
As you can see, some values of j are prime and some are not.
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: When j is divided by 2, the remainder is 1.
Possible values of j: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, ...
As you can see, some values of j are prime and some are not.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined:
From statement 1: j could equal 4, 7, 10, 13, 16, 19, 22, 25...
From statement 2: j could equal 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, ...

Some numbers that satisfy both statements are 7, 13, 19, 25, . . .
Once again, some values of j are prime and some are not.
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer = E

Cheers,
Brent